WOLFRAM NOTEBOOK

Band Structure of Graphene

​
t
1
f
AA
0
f
BB
0
Graphene consists of a single layer of carbon atoms arranged in a honeycomb lattice, with lattice constant
a=1.42Å
. This type of lattice structure has two atoms as the bases (
A
and
B
, say). In this Demonstration, the band structure of graphene is shown, within the tight-binding model. The conduction and the valence bands touch each other at six points (arranged in a hexagonal pattern) in momentum space, known as Dirac points. Three out of these six points are called
K
points and the other three are called
K'
points. In each set, the points are one reciprocal lattice vector apart. The band touching at the Dirac points gives rise to energy degeneracy at these points. The dispersion around them is found to be linear.
This Demonstration shows how the band structure changes near the Dirac points with changes in certain hopping parameters:
t
for an electron hopping from an atom at site
A
to another at site
B
f
AA
for an electron hopping from an atom at site
A
to another atom at site
A
f
BB
for an electron hopping from an atom at site
B
to another atom at site
B
In the result,
E
+
corresponds to the conduction band (in orange) and
E
-
corresponds to the valence band (in blue).

Details

The tight-binding Hamiltonian for graphene can be written as:
ℋ
=
∑
k
ψ
k
k
ψ
k
,
where
k
=
0
f
k
f
k
0
and
ψ
k
=
A
k
B
k
.
Here,
A
k
and
B
k
denote the annihilation operators for the sites
A
and
B
, respectively, in the basis of the hexagonal lattice of graphene.
f
k
is known as the hopping term, representing the hopping of electrons from atoms at site
A
to atoms at site
B
:
f
k
=-t
∑
δ
expi
k
,
δ
.
Note that
t
is nonzero only for nearest neighbor atoms, and 0 otherwise, while
δ
denotes the position vectors of the three nearest atoms in the lattice.
After diagonalizing
k
,
k
=
E
+
k
0
0
E
-
k
,
where
\b
E
±
k
=±t
3+2cos
3
k
y
a+4cos
3
k
y
a
2
cos
3
k
x
a
2
.
The expression for
E
±
k
is plotted against
k
x
and
k
y
to show the band structure of graphene. The lower band, colored blue, is the valence band, filled with electrons, while the upper conduction band is devoid of electrons. As a consequence of this type of band structure, graphene acts as a semi-metal, with its Fermi energy at
E=0
, where the conduction and valence bands meet. There are six such points in a hexagonal pattern in momentum space, called Dirac points. Near these Dirac points, the dispersion is linear and the bands are cone shaped, which is why they are known as Dirac cones.
Until now, hopping of electrons to atoms present in the same basis site (that is from site
A
to
A
or from
B
to
B
) was restricted, but if this restriction is done away with, the following changes to the previous equations are obtained:
k
=
f
AA
f
k
f
k
f
BB
.
Then after diagonalization of
k
,
k
=
E
+
k
0
0
E
-
k
,
where
E
±
k
=
1
2
f
AA
+
f
BB
±
2
(
f
AA
-
f
BB
)
+4
2
f
k
.​
This opens up gaps at the Dirac points, hence breaking the degeneracy. The band gap is given by
ΔE
k
=
2
(
f
AA
-
f
BB
)
+4
2
f
k
.
At Dirac points,
f
k
=0
, so the band gap at the Dirac points is
Δ
E
K,K'
=
f
AA
-
f
BB
.
This Demonstration shows how the band structure of graphene changes with
t
,
f
AA
and
f
BB
.
Snapshot 1: band structure with
t=1
and only
A
–
B
hopping allowed; in this case, the conduction and valence bands touch at the Dirac points
Snapshots 2, 3: the gap is opened up by increasing
f
AA
and
f
BB

References

[1] C. Kittel, Introduction to Solid State Physics, 7th ed., New York: Wiley, 1996.

External Links

Permanent Citation

Sarbani Chatterjee, Sohini Chatterjee
​
​"Band Structure of Graphene"​
​http://demonstrations.wolfram.com/BandStructureOfGraphene/​
​Wolfram Demonstrations Project​
​Published: March 20, 2020
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